Small cases lie — page 8
Geometry
Three circles that touch and are not the largest
In 1803 Gian Francesco Malfatti asked how to cut three round columns from a triangular block of marble with as little waste as possible, and answered with three circles each touching the other two and two sides. The circles exist in every triangle and are found by solving three equations. They are never the answer to his question: the plain greedy rule — the inscribed circle first, then the largest circle that still fits, twice — always does better, by 1.4 per cent on the equilateral triangle and by almost double on a thin one.
Number
Three unit fractions for every four over n
Two neighbouring fractions in the tree always differ by a unit fraction, so stepping down the tree writes any fraction as a sum of them — four over n in at most four steps. Erdős and Straus asked in 1948 whether three always suffice. Three identities settle every n except those leaving remainder 1 on division by 24, a search settles every one anyone has tried, and nobody has a proof.
Applied
A vote worth nothing until it was outnumbered
From 1958 to 1973 Luxembourg held one vote of seventeen in the Council of the European Communities and could never once change an outcome. When the Council grew, Luxembourg's share of the votes fell and its share of the power rose from nothing. Power is not a quantity a member holds; it is a property of the whole assembly, and changing the assembly moves it in directions nobody would guess.
Applied
A third kind of member
Two sides always have a stable matching, and the proof is a procedure. Add a third kind of member — capitals who rank numbers, numbers who rank letters, letters who rank capitals — and every market small enough to check still has a stable grouping. Ten million markets of three have at least ten each. Nobody can prove it continues, because the structure the two-sided proof stands on is gone.
Discrete
Almost every tree can be turned over
Cayley's n^(n−2) counts trees with labels on their points. Take the labels off and the count has no formula, because a symmetric shape absorbs labellings: the star on seven points can be labelled only seven ways, the one asymmetric shape 5,040. The bookkeeping that reconciles the two counts says something unexpected — almost every tree, labelled or not, can be turned over onto itself, where almost every graph cannot.
Number
Abundant, and still not a sum of its parts
Seventy's proper divisors add to seventy-four, more than seventy itself — and yet no selection of them adds to exactly seventy. The reason is that the excess, four, cannot be made from the small divisors one, two and five. Numbers like seventy are called weird; there are thirty-six below twenty thousand, every one of them even, and nobody knows whether an odd one exists.
Applied
Which vote finds the truth more often
A court that must decide two premises and their conjunction can vote on the premises or on the conclusion, and the two votes sometimes disagree. Suppose there is a right answer and each judge is more often right than wrong. Then the question has an exact answer: for three judges the conclusion vote is more often right when competence is below √½ and the premise vote above it, and in a large court the premise vote wins at any competence — while the conclusion vote turns out to be a built-in standard of proof.
Probability
A wait that ends and has no average
Draw a number, then keep drawing until one beats it. Half the time the very next draw does. The wait is certain to end, it is usually short, and its average is infinite — and the chance of still waiting after n draws is exactly 1/(n + 1), whatever the numbers are drawn from, because symmetry is the only thing the calculation uses.
Probability
Two-thirds of a step past the line
Add random numbers between nought and one until the total passes one, and it takes e of them on average. Ask the same about passing any t and the answer is an exact finite sum that settles, within a few units, onto the line 2t + 2/3. The slope is one over the average step. The two-thirds is the interesting part: it is what the one step that crosses the line charges for being chosen by its length, and it is set entirely by how much the steps vary.
Algebra
Nineteen copies and no more
Two of the icosahedron's rotations, chosen at random, generate all sixty of them nineteen times in thirty. Philip Hall found the number in 1936 by inclusion and exclusion over every subgroup at once, weighted by the Möbius function of the subgroups' order — and the same count says that nineteen copies of the group, side by side, can still be generated by two elements, while twenty cannot.
Analysis
Seven times half of π, and then not
The area under sin x / x, out to infinity, is π/2. Multiply in sin(x/3)/(x/3) and the area is still π/2; multiply in the waves for 5, 7, 9, 11 and 13 and it is still π/2, exactly. Add the wave for 15 and the area falls short — by about 2.3 × 10⁻¹¹. Nothing about the curves changes at the eighth factor. What changes is that 1/3 + 1/5 + … + 1/15 has just passed 1.
Applied
Every premise makes the verdict vote stricter
A court that must find several things true before it finds liability can vote on each of them or on the verdict. With two elements, judges right seven times in ten are enough for either. With five, the verdict vote needs judges right eighty-seven times in a hundred, or a larger court makes it worse — while the premise vote still needs only better than even. And when any one ground is enough instead of all, the two procedures swap their failures.
Geometry
What a junction can save
Join a set of towns by roads that may meet only at towns, and the shortest such network is easy to find. Allow junctions anywhere and the network gets shorter — but never, it is conjectured, by more than 13.4%, the saving the Fermat point makes on an equilateral triangle. Computed exactly on 1,304 random sets of up to seven towns, the average saving is about 3%. An adversarial search drives it down to the bound and no further, and a hexagon with its centre attains it exactly. The proof of the bound that was accepted for twenty years has a gap.
![Primes in five thousand thin sectors. 581517 primes in (2e7, 4e7] in 4967 sectors; mean 117.08, variance ratio 0.874, 2 empty.](/f/primes-in-five-thousand-thin-sectors-90657fc7.svg)
Number
Sectors that shrink with the primes
Hecke proved that the angles of the primes a² + b² are evenly spread, so every fixed sector eventually gets its share. Let the sector shrink as the primes grow, to width X^(−α), and the question is open beyond small α. Measured on the 581,517 such primes between twenty and forty million, wide shrinking sectors fluctuate far less than chance and thin ones as much as chance — and the thin sectors that go empty are not scattered: they gather beside the directions of small rational slope, most of all slopes like 1/3 and 1/1 whose two terms are both odd.
Probability
The longest run has no limit
Toss a fair coin n times and the longest run of heads is about log₂ n. The average settles: it is log₂ n − 0.667 to within a few millionths by a million tosses. The distribution does not. Because the run is a whole number and log₂ n is not, the chance of each value swings round the same cycle once every doubling of n, for ever, and computing it exactly to a million tosses shows the swing as large at the end as at the start.
Probability
A share that depends on the average
What share of the whole numbers begin with the digit 1? Counted up to N, the answer swings between one ninth and five ninths for ever, and averaging the swing over N, even three times over, narrows it without removing it. Weight each number n by 1/n instead and the share settles at log₁₀ 2 — Benford's value — and so does every other sensible weighting that counts each decade alike. The primes behave the same way. Benford's law for the counting numbers is not a fact about them; it is a fact about how they are averaged.
Algebra
Twenty-seven million cubics, sorted by their symmetry
Write down every cubic x³ + ax² + bx + c with coefficients up to 150 and sort each by its Galois group. 98.7% have all six symmetries of three roots. The rest are reducible at a rate of fifteen in every H², or cyclic at five in every H to the three-halves — and the cyclic ones turn up four times as often as chance would allow, because discriminants are not random numbers.
Algebra
The quartics whose roots stay paired
A quartic can have five different Galois groups, and in a box of nearly fourteen million quartics they do not come in order of size. The eight-element group D₄ outnumbers the twelve-element A₄ thirty-five to one. Every even quartic has a smaller group than S₄, and moving one step off that plane restores the full group almost everywhere.
Analysis
A sum of factorials that converges at every prime
1 + 1 + 2 + 6 + 24 + 120 + … grows faster than any geometric series and has no sum in the real numbers. Measure size by divisibility instead and the same series converges at every prime at once, to a different number each time. Whether any of those numbers ends in zero is a question Đuro Kurepa asked in 1971, and a search through every prime below 100,000 says no while a coin-toss model says it should have said yes about twice.
Discrete
Log-concave with nothing to make it so
The chromatic polynomial of a graph counts its colourings, and its coefficients rise to one peak and fall, each squared at least the product of its neighbours. For a real-rooted polynomial that would be automatic. But by nine vertices fewer than one graph in twenty has only real chromatic roots, and the coefficients stay log-concave anyway — on every graph tested, and, by June Huh's theorem of 2012, on every graph there is.
Probability
The longest thread between two random strings
Write down two random strings of a thousand bits each and find the longest sequence that can be read in both, skipping freely. It is about 81% of each, and the exact share — the Chvátal–Sankoff constant — has resisted computation for fifty years. Its fluctuations are a smaller mystery with a sharper edge: the variance is known to be at most n/2, and over every length that can be computed it grows like n to the power 0.7, with nobody able to say what it does after that.
Number
When Euclid's number is itself prime
Multiply the primes up to p and add one. Euclid needed only that the result has a prime factor beyond p; it is itself prime for p = 2, 3, 5, 7 and 11, and then not again until 31, and then not until 379. Up to 1,100 there are nine such primes of each sign, against thirteen that a careful estimate expects. The estimate predicts infinitely many and grows by about 1.8 every time x is multiplied by e; nobody can prove there is even one more.
Number
The numbers that fool every base
561 = 3 · 11 · 17 passes Fermat's test for every base it shares no factor with, and so do 1,105, 1,729 and 102 more numbers below ten million. They were proved infinite only in 1994. Counted to 10²¹ there are about twenty million, growing like x to a power that has crept from 0.21 to 0.35 — just past the bound the proofs guarantee and nowhere near the power of one that Erdős's argument predicts.
Dynamics
Stable cycles a hair from Hénon's chaos
Nobody knows whether the Hénon map at a = 1.4 is chaotic, because a stable cycle in a window too thin to see would look exactly the same. So hunt the windows: a stable orbit of period 19 at a = 1.4001616, proved to exist and to attract by interval arithmetic, in a window 15 hundred-millionths wide and 0.00016 from the classical value — and nothing nearer, down to a spacing of a ten-billionth.